Find CDF, Variance and find the parameter to given density.

Question

Let $X$ be a random variable with density $g(x)=cx^4\mathbf{1}_{[0,6]}$, where $c$ is an appropriate parameter. Then:

$(A)$ Find cumulative distribution function of random variable $X$ ;

$(B)$ Calculate the Variance of $X$ ;

$(C)$ parameter $c$ is equal to $c=\frac{6}{6^4}$ (prove or disprove it) ;

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$(A)$ $F_X(t)=P(X\leq t)=\int_{-\infty}^t cx^4\mathbf{1}_{[0,6]}dx=\int_{0}^t cx^4 dx=\frac{c}{5}t^5$

$(B)$ $Var(X)=EX^2-(EX)^2=...=\frac{c}76^7-(\frac{c}6)^26^{12}$

$(C)$ ?

So my request for this post is, please check the above calculations and formulas in particular, and send me any tips according to $(C)$, because I have no idea how to do it. Any help will be appreciated.

Answer 1

$c$ is characterized by the following equation having to hold for a pdf:

$$\int_{-\infty}^{\infty}cx^4\mathbf{1}_{[0,6]}\ dx=c\int_0^6x^4\ dx=1.$$

Then $$\int_0^6x^4\ dx=\frac156^5.$$

So, $$c=\frac5{6^5}.$$ By this, $c=\frac{6}{6^4}$ has been didproved.